Stream,current refer to a steady flow. In this use they are interchangeable. In the sense of running water, however, a stream is a flow that may be as small as a brook or as large as a river: A number of streams have their sources in mountains.Current refers to the most rapidly moving part of the stream: This river has a swift current.

stream

n.

Old English stream "a course of water," from Proto-Germanic *straumaz (cf. Old Saxon strom, Old Norse straumr, Danish strøm, Swedish ström, Norwegian straum, Old Frisian stram, Dutch stroom, Old High German stroum, German Strom "current, river"), from PIE root *sreu- "flow" (see rheum). Meaning "current in the sea" (e.g. Gulf Stream) is recorded from late 14c. Stream of consciousness in lit crit first recorded 1931, originally in psychology (1855).